`lim_(theta->(pi/2)) (1-sin(theta))/(csc(theta))` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s...

`lim_(theta->(pi/2)) (1-sin(theta))/(csc(theta))` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.

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Chapter 4, 4.4 - Problem 16 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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lemjay | High School Teacher | (Level 3) Senior Educator

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`lim_(theta->pi/2) (1-sin theta)/csc theta`

The function `(1-sin theta)/csc theta` is defined at `theta=pi/2` . So to take its limit, there is no need to apply the L'Hospital's Rule.

Instead, proceed to plug-in `theta=pi/2` .

`= (1-sin (pi/2))/csc(pi/2)`

`= (1-1)/1`

`=0/1`

`=0`

Therefore,   `lim_(theta->pi/2) (1-sintheta)/csc theta=0.`

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